Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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How to find the smallest set of generating elements in a group?

Is there a systematic procedure for finding the smallest set of generating elements of a finite group?
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24 views

Relation between permutation group and algebraic equation. [duplicate]

What kind of relation do algebraic equeation and permutation group have? For example, $Z^n -1=0$ is related to a cyclic group $C_n$. Is there anything else in this kind problem? I have read about ...
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1answer
33 views

Permutation of Disjoint Sets of a Symmetric Group

Problem Description: Consider a symmetric group $S_n$ acting on $n$ objects. We partition $S_n$ into two sets $A, B$ such that $A \cap B= \emptyset$ and $A \cup B = S_n$. In other words, $S_n$ is ...
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1answer
45 views

How string isomorphism is used in graph isomorphism?

Graph isomorphism is a special case of string isomorphism problem. In the paper of Graph Isomorphism in Quasipolynomial Time, the relation has been shown. Let, two strings $x,y$ are associated with ...
3
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1answer
25 views

Prove that L(x) = (x - 1)/p is a discrete logarithm function in a group.

I have the following problem:$$$$ Let $p$ be a prime number and $G$ be a set of all $x\in \mathbb{Z}_{p^2}$, such that $x \equiv 1 \pmod{p}$. Prove that: $G$ is a multiplicative group (regarding ...
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1answer
25 views

String isomorphism definition: Is it for any arbitrary group?

Scott Aaronson's blog, I find the description of string isomorphism as- you’re given two strings $x$ and $y$ over some finite alphabet, as well as the generators of a group $G$ of permutations ...
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23 views

Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
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1answer
26 views

Supersolvable and pronormal subgroups

Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....
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Is the convolution algebra a *-algebra?

Let $G$ be a finite abelian group with $n$ elements. Consider the convolution algebra $C^*(G) \subset l^2(G)$, with multiplication: $$(a * b )(g) = \frac{1}{n}\sum_{x\in G} a(x)b(g-x)$$ Is there a ...
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4answers
109 views

Infinite groups with all elements of order 2?

If G is a group such that $a^2 =e$ for all $a \in G$, where $e$ is the identity element in $G$, then $G$ is finite. This question can be proved false if we can get a group of infinite order with ...
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1answer
149 views

Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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0answers
35 views

Group action with two normal subgroups which induce same block system

So awhile back I asked this question here on stack exchange: Normal subgroup $H$ of $G$ with same orbits of action on $X$. At the time I wasn't quite sure what I was really wanting to know about ...
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0answers
21 views

Irreducible representation of $1$-transposition groups

I would like to know the theory of irreducible representation of $1$-transposition groups. Could anyone provide me a pointer from where I can proceed?
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0answers
29 views

Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
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0answers
23 views

Automorphism groups of partially cycle graphs

I define partially cycle graphs as follows. If we add the same subgraph to $n-k$ vertices of an $n$-vertex cycle graph, where $1\le k < n$, we create a partially cycle graph. Here are a few ...
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2answers
97 views

Groups and rings of order $p^2$.

Up to isomorphism there are exactly two abelian groups of order $p^2$. there are exactly two groups of order $p^2$. there are exactly two commutative rings of order $p^2$. there is exactly one ...
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0answers
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Let $G = \mathbb{Z}_5 \times A_5$. if $H$ be a subgroup of $G$ of order $5$, then $H$ is weakly $s$-permutably embedded in $G$.

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. $H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some $s$-...
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0answers
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If a finite group G is a normal subgroup of the automorphism group of any connected Cayley graph thereof, is it realizable over Q?

The question is in the title: let $G$ be a finite group, and assume it is a normal subgroup of the automorphism group of any of its connected Cayley graphs. Does this imply that $G$ is the Galois ...
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0answers
101 views

How many groups of order $2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2\cdot 13^2$ exist?

The calculation of the number of groups of order $$2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2$$ (result $81883$) takes already two hours with GAP. So, the calculation of the number of groups of order ...
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2answers
62 views

Is a formula for $gnu(2pq^2)$ known, where $q=2p+1\ $?

Let $p$ be an odd prime such that $q:=2p+1$ is also prime. Denote $g(p):=gnu(2pq^2)$ = number of groups of order $2pq^2$ upto isomorphy. The following table shows the first few values ...
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0answers
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If a conjugacy class intersect with its centralizer what can be said about its elements?

Suppose that $G$ is a finite group and let $x\in G$. If $y\in x^{G}\cap C_{G}(x)$, what can be said about the relationship of $x$ and $y$, or anything about $x$?
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1answer
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Lattice of Subgroups and Automorphisms

So I have a rather interesting question that came up in some independent research I have been doing on finite groups of small order. I was looking at their (full) subgroup lattices, which included the ...
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1answer
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Counting number of cosets

Let $G = \big(\mathbb{Z}/n\mathbb{Z})^*$, that is the multiplicative group modulo $n$. For some $d$ coprime to $n$, let $H$ be a subgroup of $G$ generated by $d$. As $G$ is abelian, $H$ is normal in $...
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(B,N) pair and Steinberg idempotent

Let $q=p^f$ where $p$ is prime and $G$ be a finite group with a $(B,N)−$pair ($T=B\cap N$ and $W=N/T$), and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Define $$e=\dfrac{1}{[G:U]}\...
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1answer
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(B,N) pair and normal subgroup

I am trying to prove the following: Let $G$ be a finite group with a $(B,N)-$pair and assume that $B=UT$ with $U\triangleleft B$ and $U\cap T=1$. Let $\widetilde{G}\triangleleft G$ such that $U\le \...
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1answer
69 views

Condition that for a given set of numbers and given divisor all finite sums from this set contain all possible remainders

Given $q \in \mathbb{N}$ and ${a_1, a_2, ...}$ where each $a_j \in \mathbb{N} \cup{\{0\}}$ define $A_p=$ {set of all finite sums of $\{a_1 ... a_p\}$ such that each $a_j$ will appear either $1$ or $0$ ...
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1answer
74 views

Is the notion of Stabilizer of a subset A of a group G absurd?

Is the notion of Stabilizer of a subset,A of a group G is absurd? I don't know whether this makes sense or not,but for curiosity i want to know view of experts. UPDATE i'm dealing with ...
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2answers
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Showing: if G acts on A by conjugation then the stabilizer of A in G is the Normalizer of A in G.

This is a theorem from Dummit & Foote text- The number of conjugates of a subset $ A$ in a group $G$ is the index of the normalizer of $A$,$\vert G:N_G(A) \vert$. The highlighted text is a ...
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1answer
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Not getting how to prove reverse hypothesis.

This is a theorem from Dummit & Foote text- Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by $a \sim b$ iff $a=g.b$ for some $g \in G$ is an ...
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Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
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1answer
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Condition in a theorem of Hall

There is a well-celebrated theorem of Hall, which characterizes solvable groups according to the existence of Hall-$\pi$ subgroups. In this theorem, I was wondering whether it can be stated in a ...
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Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +)

Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +) where k=phi(m) or to Za x Zb x Zc x....where abc..=k with the right combination of a,b,c... ?
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1answer
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generators of $\mathbb{Z}_p^*$ are all the elements in $\mathbb{Z}_p^*$?

I know that a finite group with a prime number of elements is cyclic and every element in the group is a generator for the group. Thinking about $(\mathbb{Z}_p^*, \cdot)$ I thought that the order of ...
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3answers
35 views

Group theory finding proper subgroup [closed]

The smallest order for a group to have a non-abelian proper subgroup? I am confused how shall I proceed pls help Thanks in advance
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1answer
36 views

Is there an onto homomorphism $S_4\to S_3$ [duplicate]

Prove or disprove that there is an onto homomorphism from $S_4\to S_3$ where $S_n$ is the symetric group of order $n!$. after long time of searching, I finally success but i just manually tried to ...
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1answer
29 views

Need help for the example on conjugation.

This is an example from Dummit & Foote text, i've some queries in this- If $|G|>1$, then unlike action by left multiplication,$G$ does not act tranistively on itself by conjugation because {1}...
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0answers
28 views

Permutations associated to a transversal and Cayley theorem

Let $G$ be a finite group with $H\le G$ and $T$ a right transversal of $H$ in $G$. $G$ acts on itself by left multiplication and so we can consider $G\le \mathfrak{S}_G$. Let $g\in G$. The permutation ...
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0answers
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Fixed Point of Automorphism group of a Cyclic group Z2XZ2^2 I need the command on GAP

Dear Mathematics Stack Exchange, I have a problem that how to write a command in GAP the automorphism group of finite abelian group and their fixed points. Let Z_pXZ_p2 be cyclic group where p is ...
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0answers
62 views

How can we show $gnu(8892)=gnu(9324)$ by hand?

With GAP it can be verified immediately that there are $303$ groups of order $8892=2^2\cdot 3^2\cdot 13\cdot19$ and also $303$ groups of order $9324=2^2\cdot3^2\cdot 7\cdot 37$. I do not expect that ...
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1answer
19 views

Show how (0,(12)) and (1,(12)) are in different conjugacy classes.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
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Class-equation of $\mathbb Z_2$ $\oplus$ $S_3$.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
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0answers
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Intersection of the kernel of the irreducible characters determinants

Let $G$ be a finite group. It is easy to show that $G'\le \bigcap_{\chi\in Irr(G)}Kerdet\chi$. Is there equality ? This question arises from the remarkable equalities $\bigcap_{\chi\in Irr(G)}Ker\...
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0answers
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Determinant of a character

let two characters $\chi$ and $\vartheta$ of a finite group $G$ (assumed to be non-null). Let $\mathfrak{X}$ and $\mathfrak{Y}$ be representations of $G$ affording respectively $\chi$ and $\vartheta$ ...
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2answers
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Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
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2answers
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Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
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1answer
57 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
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1answer
47 views

Problems related to conjugacy classes and their sizes.

Is $Z( G_1 \oplus G_2 \oplus G_3 \oplus\cdots\oplus G_n) = Z(G_1) \oplus Z (G_2) \oplus Z(G_3) \oplus \cdots \oplus Z$ $(G_n)$ true, where $ G_1, G_2, G_3 \cdots G_n$ are finite groups?$Z$,here refers ...
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2answers
47 views

Subgroups of finite abelian groups.

For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example. I am aware of isomorphism theorems ...
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0answers
81 views

Almost all finite groups have order $2^n$?

This might be a stupid question, but here it goes: Is anything known about, whether: $$\lim_{n\to \infty} \frac{\#\{\text{Groups of order }2^n\}}{\#\{\text{Groups of order} \leq 2^n\}} = 1$$ (where ...
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0answers
27 views

Differences among the Group Cohomology with coefficients over a commutative ring and coefficients over an arbitrary $G$-module

Assume that $G$ is a finite group and $k$ is an arbitrary commutative ring. From the general theory we know that the group cohomology $H^{*}(G ; k)$ becomes a graded ring (with the cup product). ...